Thursday, December 19, 2013

In the suburbs of
Bangalore, in one of the numerous buildings that house research and support
facilities for nearly every major tech company in the world, scientists are
working on understanding how you spread your attention when you navigate a web
page. A few of them had gathered in a
conference room, listening as I described some of our work on how the brain
controls movements of the eye.

Gazing at
the teak conference table, high back leather chairs, and sophisticated teleconferencing
equipment, I considered the contrast: just a few streets away from this modern
world where I was giving my talk, there were goats munching on a pile of
refuse, and a small band of cows roaming happily against traffic. A little farther, in the center of the city,
there were scientists and engineers working on fundamental questions in the
Indian Institute of Science, a major university on a beautiful wooded campus
that housed, in addition to world class laboratories, large families of
monkeys, bands of wild dogs, and bats the size of crows, all living freely, and
from all indications, contently, alongside humans.

I think the most
striking difference with anywhere else that I have visited is that people here
seem to have an exceptional respect for life --- life of any form. Like most
university campuses, this one also has large, impressive trees that dot the
landscape. But here, the human roads do not prevail. Indeed, in many places the road has a large
tree in the middle of it, with a trunk marked with a few reflectors, and the
cars simply go around it. At our university guest house, a sprawling
hotel-like structure, there are a few places where the hallway turns at a
strange angle. Looking closer, I see
that the building is bending around an old tree, and not the other way
around. This co-existence is on display
with the wildlife that lives alongside us.
The faculty housing is in a wooded area, where monkeys also raise their
families. One morning, as we ate
breakfast at the guest house, with the window open to let in the cool breeze, a
family of macaque monkeys came to visit.
The mama-monkey took a piece of papaya from a table, and went over and
fed her babies.

The weather is mild
and pleasant; a pleasure to step outside and feel the sun and smell the trees.
But the university is an oasis. The peace and quiet of the grounds
are in stark contrast to the outside world. As we step beyond the gates,
we leave "jungle book" and enter the human world; with its crushing
traffic of cars, motorized three-wheel rickshaws, and scooters, all
communicating in the machine-made language of horns.

The human languages
are myriad in India, but the main language, at least here in the south, is
English. The students tell me that they rely on English to talk to each
other because each comes from a different part of India, with its own languages,
and English is the only common tongue.

The diversity of
languages is complemented with the diversity of faiths. In the mornings, I hear the Muslim call to
prayer before sunrise, and then a few hours later, I see the Hindu temple as I
walk to the university conference center.
On the steps of the center there is a familiar scene, one of the wild
dogs napping in the sun.

Tuesday, December 3, 2013

A few months back, my administrative assistant was offered a
wonderful new job and as a consequence, the department hired a replacement. The new assistant is a capable, hardworking young
lady. A few days ago I noticed that she
tends to use the word /axe/, instead of /ask/.

I had heard this usage a number of times in Baltimore, particularly
among African-Americans. I wondered, is
this a mispronunciation? Perhaps
something like /nuclear/ vs. /nucular/?
A bit of research made me understand that /axe/ has a long history in the
English language, and is not a mispronunciation.

Oxford dictionary notes that /ask/ is the descendant of /ascian/,
which in Old English means to demand, to seek from. The alternative form of /ascian/ is /axian/,
or in short form, /axe/. Oxford notes its
use in Chaucer: "I axe, why the fyfte man Was nought housband to the
Samaritan?" (Wife's Prologue 1386), and "a man that ... cometh for to
axe him of mercy." (The Parson's Tale 1386) The book The Complete Works of Goeffrey
Chaucer includes 5 passages where the word /axing/ is used.The word /axe/ appeared in the first complete
English translation of the Bible in 1535 by Miles Coverdale, who wrote: "Axe
and it shal be giuen you," and "he axed for wrytinge tables."

According to Random House, “In American English, the /axe/
pronunciation was originally dominant in New England. The popularity of this
pronunciation faded in the North early in the 19th century as it became more
common in the South. Today the pronunciation is perceived in the US as either
Southern or African-American. /axe/ is still found frequently in the South, and
is a characteristic of some speech communities as far north as New Jersey, Pennsylvania,
Illinois and Iowa.”

So /axe/ is a regional pronunciation, somewhat similar to
the regional pronunciation variation of the word /idea/ and /idear/.

Saturday, September 21, 2013

When he walked into the room, he looked a decade younger
than 71; his face handsome, with few wrinkles.
I pulled out a chair and asked him to sit in front of a robotic
contraption that earlier that morning we had set up in an examination room at
the Clinical Research Center at MIT. He
sat calmly and avoided touching the contraption. I pulled the robotic arm toward him and asked
him to hold its handle. He grabbed the
handle and started moving it around, keeping his gaze on the handle. I asked him to look up at a monitor, where he
saw a little cursor moving around as he moved the robot’s handle. The computer displayed a target box, and he
moved the cursor into the box, at which point the computer animated it,
producing an explosion.

A smile came to his face.
He said: “You know, when I was a kid I liked to go bird hunting.” Exuberantly,
he described the birds that he hunted, the guns that he owned, and the woods
around his childhood home. He continued
doing the task, reaching while holding the robotic arm and making little
explosions. About five minutes later he
said: “You know, when I was a kid I liked to go bird hunting.” The exuberance was unabated. He had no idea that a few minutes earlier he
had told me that same story.

Memory without
awareness

The day before, with my two graduate students Kurt Thoroughman and Maurice Smith, I had packed the robot and computers in the back
of my wife’s station wagon and drove up from Baltimore to meet and examine
Henry Molaison. Henry, or as he is known
to the scientific world ‘H.M.’, had suffered from debilitating seizures. When he was 27, desperate for something that
might help, he had agreed to an experimental procedure that surgically removed
the hippocampus and amygdala from both sides of his cerebral cortex. The surgery was successful, greatly reducing his
seizures, but left him with a staggering deficit: an inability to form certain
long-term memories.

Now, in the examination room, the robotic arm that Henry was
moving started to produce forces, pushing his hand as he approached the target,
making it so that he would miss-reach and not get those explosions. But he kept practicing, and after a few
minutes, a part of his brain that was not damaged learned how to generate the
right motor commands so that his arm could compensate for those unusual
forces. Once again he got the
explosions, and once again he excitedly told me of his childhood bird hunting
days.

After about an hour of playing with the robotic arm, Henry
left for lunch and an afternoon nap. He
returned about 4 hours later. I said
hello and ask him whether he remembered meeting me and playing with the robotic
arm. He said no. I pushed the robot aside, showed him the exam
chair, and asked him to sit down. He sat
down, but then something interesting happened: rather than avoiding the
machine, the behavior of someone who has never done the task before, he grabbed
its handle, brought it toward him, looked at the video monitor, and started to
move the cursor toward the target.

He had no awareness that he had seen me before, or that he
had played with this robotic arm only hours earlier. Yet, that experience had left two kinds of memories
in his brain: the memory of how to use
the tool, and the memory that associated the sight of the tool, and the act of
moving it, with a rewarding outcome.

He was not aware of it, but the sight of the strange tool,
the robotic arm, was sufficient for him to want to hold it and move it around
so that he could chase targets and get explosions. Would he have voluntarily reached for the
robot if, while playing with it, he had not had a pleasurable experience,
recalling those childhood memories? Probably
not. In 1911, Edouard Claparede, a physician
in Geneva, described an amnesic woman much like Henry. Claparede had wanted to test her memory, so
he played a small joke on her: when he reached his hand out to shake hers, he
had hidden a small tack in his palm.
When the patient shook his hand, she felt the sharp tack. The next day when Claparede had approached
her, she could not recall having seen him before. However, when he reached out to shake her
hand, she pulled away, despite being unable to say why she did not want to
shake his hand.

Henry could not remember the episode of having played with
the robotic arm, but that act left a memory, associating the robot with a
rewarding outcome. The part of his brain
that learned this value association was exhibiting its knowledge by reaching
out to the robot and moving it in search of a target to explode.

In addition to this value-action association, he also had a
memory of how to use the contraption.
Those forces that he had practiced to overcome had left a different kind
of memory: much like picking up a coke can that you expect to be full but is
empty, Henry’s movements on this revisit had the ‘after-effects’ of the earlier
experience. The robot was not producing
any forces, but he moved it as if he was expecting it to be producing those
earlier forces. When the forces were
re-introduced, he moved the robot skillfully.

He did not have the ability to form memories of episodes of
his life, but these two other intact forms of memory served him well. For example, as he aged, he developed
osteoporosis and required a walker to keep physically active. With practice, he learned to use the walker skillfully. Importantly, he would use the walker without
being told to do so. That is, he ‘knew’
that the walker was a useful tool that helped him get around.

Permanent present
tense

What is it like to live a life with such a disability? In a recent book titled “Permanent present
tense”, Suzzane Corkin, a scientist who studied and cared for Henry for 46
years, describes his life in loving, exquisite detail.

In a photograph showing Henry with his mother at his 50th
birthday, he recognized his mother, but not himself. While attending his 35th high
school reunion, he did not recognize anyone by sight or by name. He could not recall any specific event in his
life, even events from before his operation. For example, he could not recall a single specific
Christmas gift that his father had given him.
He remembered some of the facts that he had learned from the time before
his operation, and the gist of the experienced events, but no recollection of any
specific episodes.

Henry rarely spoke of being hungry or thirsty. He never sought out food for himself; it was
simply given to him by his caregivers.
In 1981, Corkin asked him to rate his hunger from 0 (famished) to 100 (absolutely
full). He consistently gave a rating of
50, whether he had just finished eating, or was about to eat. One evening, Corkin played a small trick on
him. After Henry had finished eating and
his tray had been taken away, the kitchen staff brought him another tray, with
exactly the same meal. Henry ate the
second dinner, cleaning the plates, except for the salad. He seemed unable to express a feeling of satiety.

When we were examining him, a caregiver mentioned that Henry
rarely verbalized that anything might be wrong.
For example, if he had a tooth ache, he would rarely mention it. Only by observing that he was deviating from
his normal behavior during the day would the caregiver suspect that something was
wrong. The caregiver would then go
through a list of things to see if they could find out what may be the problem.

Corkin tested Henry’s ability to perceive pain by using a hairdryer
to project a spot of heat onto his skin.
The heat was not intense enough to burn the skin, but the idea was to
test whether Henry could feel pain.
Corkin’s results showed that Henry not only could not discriminate
normally between various levels of pain, he did not report any of the stimuli
as painful, and never withdrew his arm. It
is possible that the inability to normally perceive pain, to know hunger or
thirst, was related to his operation; perhaps it was associated with removal of
the amygdala, as Corkin suggests.

Henry lived a life without keeping memories of the events, and
without pain. His father had passed
away, and his mother, who had taken care of him for much of his life, was in a
nursing home. He kept two notes in his wallet that he had written to himself: “Dad’s gone”, “Mom’s in nursing home—is
in good health.”

Sunday, August 25, 2013

At the age of 88, my seemingly healthy and vigorous father
suddenly died. He had commanded an army,
lived through a revolution, met kings and presidents, and through it all raised
a family. And then one night, all those
memories, all those experiences, vanished.
Coming home from the funeral I realized that a library, stacked with
history books, had burned to the ground.

When we experience something, it can become a memory. But what is this memory? What is its neural substrate? Can someday memories that we store in our
brain be read out and stored in a machine?
Is there any hope that the library can be saved from the fire that
consumes us as we die?

Standard model of memory

Our model of memory today is one of synaptic
plasticity. When we experience
something, the neurons that are engaged by that experience produce electrical activity, and that
activity can alter the strength of synapses that connect them to other
neurons. The electrical activity can also result in
growth of new synapses. Together, this
altered strength of connectivity in an existing network of neurons is thought
to be the basis of memory. So in
principle, if one could measure the strength of each synapse, and model the
functional properties of each neuron, then one has a representation that
approximates the state of brain of an individual. The lifetime of memories and experience are
within this representation.

The problem, unfortunately, is that this concept of memory relies
on the assumption that neurons themselves are fixed nodes, whereas the
connections (that is, the synapses) are the changing components through which
memories are stored. This assumption, as
it turns out, is false. New neurons are
born every day, and the human brain, even in old age, adds and subtracts nodes
to the network.

Finding a neuron’s birthday

Between 1955 and 1963, there were numerous above ground tests
of nuclear weapons. With every
explosion, the amount of isotope 14C was elevated in the atmosphere. In 1963, there was a treaty that banned such
tests, and since then the atmospheric level of 14C has declined because of
uptake by plants. This uptake takes
place as 14C in the atmosphere reacts with oxygen to make CO2, which is then
taken up by plants in photosynthesis.

When we eat plants, or eat animals that feed on plants, the
14C is transferred to our body. Once
transferred to our body, 14C becomes part of the DNA of new born cells. This happens when a cell divides and makes a
copy of its chromosomes. The copying
process integrates the 14C into the newly made genome, making it so that by
looking at the concentration of 14C in a cell’s DNA, and comparing it to the
atmospheric DNA, one can tell when that cell was born.

Kristy Spalding, Jonas Frisen, and their colleagues used
this idea to find the birthday of neurons in the human brain. In their study, they examined brains of people
who had died between 2000 and 2012.
These people had had their brains preserved during autopsy, and so their
brain could be studied.

They focused their efforts on the neurons in the hippocampus
region of the brain, a location that is critical for formation of new memories. The hippocampus is the place in our brain
where we form autobiographical memories, i.e., the kind of memories that
describe places and people that we have met, events that have taken place in
our life, etc. Spalding and colleagues
asked, how old are the neurons in the hippocampus of a person who was 30 years
old when she died? You might guess,
well, the neuron is probably close to 30 years old. But that assumes that all neurons are born soon
after birth. Strikingly, Spalding and
colleagues found that the neurons were much younger than the person.

Neurons are much younger than the age of the person

The authors found that for a 20 year old, the average age of
neurons in the hippocampus was 18. For a
40 year old, the average age was 29. For
a 60 year old the average age was 37.
Remarkably, for an 80 year old, the average age of hippocampal neurons was
40!

So the average neuron in the
hippocampus of an 80 year old has been around only long enough to experience
the last 40 years. It cannot ‘remember’
anything from the first half of life, because it was not around to experience
it.

Therefore, there is substantial neurogenesis throughout life
in the hippocampus. In fact, the rate of
neurogenesis showed only a modest decline with aging. They estimated that each day, 0.004% of the
neurons in the dentate gyrus of the hippocampus die and are replaced with new
ones.

Now it is possible that neurogenesis in the hippocampus is
especially high, and other parts of the cerebral cortex may not have such a
high turn-over.But the relative youth
of the neurons in the hippocampus raises a fundamental question:what is memory if neurons are eliminated and
replaced on a daily basis?

Richard Feynman, the celebrated physicist, during a lecture
in 1955 to the National Academy of Sciences, described the basic problem:

“The radioactive phosphorus content of the cerebrum of the
rat decreases to one half in a period of two weeks. Now what does that mean? It means that phosphorus that is in the brain
of the rat, and also in mine, and yours, is not the same phosphorous as it was
two weeks ago. It means the atoms that
are in the brain are being replaced: the ones that were there before have gone
away. So what is this mind of ours: what
are these atoms with consciousness? Last
week’s potatoes! They now can remember what was going on in my mind a year ago,
a mind which has long ago been replaced.
To note that the thing I call my individuality is only a pattern or a
dance… The atoms come into my brain, dance a dance, and then go out--- there
are always new atoms, but always doing the same dance, remembering what the
dance was yesterday.”

The problem in neuroscience is to understand how to read
this dance. If we could, then in
principle it should be possible to record and preserve our experiences, so that
when we die, the library will remain standing.

Wednesday, June 26, 2013

Neurons in the brain, like any other cell in our body,
require oxygen to live. They get this
oxygen from the blood vessels that run nearby.
When there is a stroke, the cause is often a particle that has gotten
stuck in a branch of an artery, blocking the flow of blood, producing
ischemia. This loss of oxygen starts a cascade of events that culminate in the
death of the neurons that live nearby. In
the last 3 years, there have been a couple of remarkable papers from a small
laboratory in University of California Irvine that suggest a new and
non-invasive way to fight this plumbing problem.

The connection between
neurons and blood vessels

When someone touches your arm, the neurons in the arm area
of your somatosensory cortex become highly active, producing what are called action
potentials. Action potentials are the
only mechanism that neurons have to communicate with each other. Generating an action potential requires
energy, and this energy is supplied via the nutrients and oxygen that are carried by nearby blood vessels. When neurons
generate action potentials, support cells that monitor the neurons send signals
to the cells that line the blood vessels, causing the vessels to locally
enlarge. This enlargement produces an
increase in the blood volume and arrival of a greater amount of food and oxygen. Indeed, this fact is the basis of a form of
functional magnetic resonance imaging (fMRI) in which blood oxygenation levels are
imaged and act as a proxy for activity in the nearby neurons. So the neurons are in close contact
with the vessels, and the vessels are the gardeners that provide the neurons
with nutrients precisely when they need it.

Activating neurons in the hour
after a stroke

When a blood vessel is blocked, the cells in the vicinity
are deprived of their oxygen. But blood
vessels are not like branches on a tree where there is only one way to get to a
spot. Rather, they are a little like the
highway system: there are multiple ways to get to a spot. This is important because blocking a branch
of an artery need not be catastrophic if a healthy branch could enlarge and
supply some of the nutrients that are needed by the cells near the blocked branch of the artery. But how can this be done?

In 2010, Christopher Lay and colleagues at University of
California Irvine reported the results of an experiment that did just that,
find a simple way to alert the healthy blood vessels to compensate for the blocked
one. In Lay et al. (2010), the authors
first took a group of rats, anesthetized them, and then gave them a stroke in
the base of the proximal middle cerebral artery (MCA). They did this by tying a suture
around a branch of MCA that supplies blood to the area of the rat’s somatosensory
cortex which encodes sensory information from its whiskers. This stopped the blood flow to that region,
causing ischemia, and produced brain damage (called an infarct). The next day, the rats were impaired in their
ability to use their whiskers, and the somatosensory cortex showed clear signs
of neural damage.

They next took another group of rats and also gave them an
MCA stroke, but rather than just letting them lie there, during the hour after
the stroke they kept touching and moving their whisker (1sec of 5Hz deflections
of a single whisker, once every 20 seconds).
Twenty four hours after the stroke, they tested the stimulated rats and
found that the damage to the neural tissue was much less than in the
non-stimulated rats. Behavior, imaging,
and neurophysiological investigation of the stimulated rats showed that by all
measures touching the whisker seemed to have made a very significant
difference.

This positive effect
happened only if the whisker was touched in the one hour or so after the
stroke. If the same touching was done at
3 hours, the effect was to worsen the stroke.
So there was a critical one hour time window after a stroke in which touching
the body part (and presumably activating the neurons that reside in the
affected cortex) seemed to dramatically reduce the damage normally caused by
the stroke. Stimulating the neurons in
the stroke affected region seemed to provide them with a pathway to survival.

How could this have happened? Further testing showed that blood reperfusion to
the affected tissue was established via collateral flow from distal branches of
the MCA (Lay et al. 2010). This reperfusion started at stimulation onset, and
then grew gradually, reaching near normal levels at around 1.5 hours (Lay et
al. 2011). The reperfusion was absent in
the non-stimulated animals. It is possible
that stimulating the whiskers immediately after the stroke had signaled a much
larger blood vessel network than the nearby, blocked vessel. In a control experiment, if a larger network
of vessels was also blocked, then the stimulation made no difference.

One problem with these studies is that the rats were fairly
young (in human terms, in their 20s).
People at that age do not usually have a stroke, and the brain is
generally more plastic and forgiving at an early age. So Lay and colleagues repeated their
experiment in elderly rats, equivalent to around 60 year old humans (Lay et al.
2012). They found that the stimulated
elderly rats suffered an infarct that was much smaller than their control
rats. Stimulation was effective in the
elderly as well as young.

Another problem with these studies is that the rats were anesthetized during the stroke and during the stimulation. Of course, people are usually awake when they
have a stroke. Did the anesthesia play a
critical role in the unusual success of the stimulation? In a further study, the authors tried a new
anesthetic that allowed them to occlude the MCA under anesthesia, but once that
surgical procedure was completed and anesthesia removed, the animal could
return to an awake state within minutes (Lay et al. 2013). During this awake state they stimulated the
whiskers and found recovery data similar to their previous results on deeply
anesthetized animals. The stimulation,
and not the anesthesia, seemed to be the key factor.

These results are all from one laboratory, and need to be
confirmed by other labs. However, the
results are tantalizing, as they suggest a stimulation based, non-invasive
strategy during a critical period after stroke that may rescue the brain.

Tuesday, June 4, 2013

In the movie
"Promised Land", there is a scene in which the main character, a
corporate type, has been sent from New York to a small farming town to lease the land for gas
drilling (played by Matt Damon). After he arrives in town, he puts on his boots to go visit a farmer and make his proposition. The
boots are old, and not much to look at, but they are something that he has
owned since his early days in a small town in Iowa. In
more ways than one, the boots are the only authentic thing about him.

It made me think of my
old boots. When I was 14, my first winter in America was approaching, and
my American mom took me to the store to get some winter boots. She, being
a gentle soul, let me choose the one that I liked. When my American dad saw them, he said that
they were fine work boots, but they had no insulation and would not really do
for the winter. They were about ready to take them back, but I hemmed and
hawed and said that I liked them and would be fine with them in the snow. And so they let me keep them.

And indeed we were
fine together. With some bees wax, I made them waterproof and then took
them on my little adventures: a week long hiking trip through the
North Cascade Wilderness (where a bear tried to tear down the bag of food that
we had hung from a tree), bucking bales on a hay farm (where I lasted only a
single day, as the bales were 60+ pounds and I weighed barely twice as much), and
oh so many fishing trips (where despite the loads of fish, we fried hot dogs for
dinner).

This afternoon, after
coming home from lab, I went down to the basement, pulled the boots out of the
shoe rack, and put them on to do some lawn edging. They are as
comfortable as a pair of slippers, though aged a bit. They are the oldest thing
that I own, and still use.

Monday, May 20, 2013

In 1974, the Journal of Applied Behavior Analysis published a most unusual manuscript. The journal received the manuscript on 25 October 1973, and published it without revision. The manuscript contained not a single word of text, except for the title, name of the author, his affiliation, the subtitle "References", and a brief acknowledgement. There were no equations, no figures, and no references. Essentially, the manuscript was a blank page, authored by Dennis Upper of Veteran's Administration Hospital of Brockton, Massachusetts. When the manuscript was published, the journal also published a reviewer's comments. Here is what the reviewer had to say: "I have studied this manuscript very carefully with lemon juice and X-rays and have not detected a single flaw in either design or writing style. I suggest it be published without revision. Clearly it is the most concise manuscript I have ever seen --- yet is contains sufficient detail to allow other investigators to replicate Dr. Upper's failure. In comparison with the other manuscripts I get from you containing all that complicated detail, this one was a pleasure to examine. Surely we can find a place for this paper in the Journal --- perhaps on the edge of a blank page."

Saturday, April 6, 2013

Some years back I sat with my son as he did his math homework.
Looking over his shoulder, I saw that he was working on the function

After he finished plotting it, I thought, let’s try another one,

I began by plotting the function for various integers of x and got the black
dots in the graph below. Then, naively, I
connected the dots:

Looking
at the plot, I realized that this could not possibly be right. How could this function cross the
x-axis? That would mean that there are
some values of x for which f(x) = 0. But there were no such values.After all, you could
not raise a number to a power and get zero.So what’s going on?

What is going on is that our function

spends most of its time outside of my piece of paper, in an ‘imaginary’ world
that lies above and below the plane of my paper. The points that I had plotted in the figure (the
black dots) are the points that I can see when this function crosses the plane
of the paper. The rest of the time the
function is outside my plane. Here is
what our function really looks like:

In the above figure, the plane of the paper is colored
green. Our function is like a
cork-screw, winding itself around the x-axis.
I wondered, how did humans discover that in addition to the “real” world
(the plane of my paper), there must also exist an “imaginary” world? What was the origin of the idea of imaginary
numbers?

In a wonderful little book called “An imaginary
tale: the story of square root of -1", Paul Nahin recounts the journey.
Surprisingly, the discovery has little to do with quadratic equations,
and everything to do with cubics.

Roots of equations

In the 16th century (and for a
century or two after that), mathematicians were very much concerned with
geometric meaning of equations. So if
you asked one what is the root of the following equation

he
or she would think about it in terms of the function

and
ask where this function crosses the x-axis.
Here is what this function looks like:

Our
quadratic function never crosses the x-axis, and so our 16th century
mathematician would respond by saying that the equation

is
impossible because

never
touches the x-axis.That would be the
end of the conversation.Indeed, as Nahin
explains, this is why the origin of imaginary numbers did not start with
quadratic equations.Rather,
“impossible” numbers like

had
their origin in cubic equations.

Depressed cubics

Scipione del Ferro was a 16th century Italian
mathematician working on cubic equations of the form:

(1)

These
are called depressed cubics because they are missing the quadratic x term. His objective was to find the roots of this
equation, which translates into finding the value or values of x for which this
equation is true. This means finding the
value of x for which the function

crosses the x-axis. A
cubic function will always have at least one location at which it will cross
the x-axis, so del Ferro knew that there must exist at least one value of x for
which this equation is true.

He started by assuming that the solution could
be written as the sum of two number, u and v:

If we put this into our cubic equation we
get:

(2)

Expanding
it we have:
(3)We
can pick u and v arbitrarily (as long as x = u + v), and so del Ferro picked u and v such that

This implies that the second term in Eq. (3) is zero, and so
we have:

(4)

To
solve the above equation set

and so we have

Del
Ferro knew how to solve quadratic equations.
We have

which
means that:

(5)

From
Eq. (4) we had

and so

(6)

The
way to understand Eqs. (5) and (6) is as follows: u can take on two values, one
given by the plus term, and the other given by the minus term. When u is given by the plus term, v is given
by the minus term, and so on. Now if we
write the solution x = u + v, we end up with the expression:

(8)

When
p and q are positive the right side of Eq. (8) will become the third root of a
negative number, which can be uncomfortable to deal with, and so let us
re-write it by noting that

Using this we can re-write Eq. (8) as:

(9)

del Ferro had found a solution to a cubic, something that
had eluded man for 2000 years, ever since Babylonian times.

This was a remarkable achievement
indeed. However, del Ferro knew that in
his equation lied a deep mystery: when p and q were both positive his equation
gave the correct answer, but when one or the other was negative, his equation
gave an impossible answer. He did not
know why this formula seemed to fail in some cases. This, it turns out, is the key mystery that
led to discovery of imaginary numbers.

The impossible equation

Consider the cubic equation

When
we plot this equation, we have:

The equation crosses the x-axis at x=2, and so 2 is one of
the roots of this equation (indeed, 2 is the only real root). Using del Ferro’s formula (Eq. 9) and a
calculator we find that the rather hairy calculation produces an answer that
is, remarkably, exactly 2. So far so
good.

Next, let us try the cubic

When
we plot this equation, we have:

We see that x=4 is a solution. In fact, our cubic crosses the x-axis three
times, and one of those times is at x=4 (this cubic has three real solutions). But now let us try del Ferro’s formula. From Eq. (8) we have:

(10) But if del Ferro’s formula is correct, then the following
must be true:
(11)

And so we arrive at the mystery: we know that x=4 is a
solution to this cubic, and we know that del Ferro’s formula is correct. Yet, when we use it, we get what appears to
be an impossible equation (Eq. 11): we have two instances of a square root of a negative
number, which at del Ferro’s time were thought to be meaningless, and yet when
these two numbers add, they produce a real number! How could that be?

It took another 50 years of thinking, and the result was a
book entitled Algebra (1572), by Rafael Bombelli, a mathematician that received
no college education. He was the first
to see that Eq. (11) required existence of a whole new set of numbers, called
imaginary numbers.

He proposed that perhaps Eq. (11) is true because each of
the third roots produce something that is partly real, and partly imaginary,
and the sum causes the two imaginary parts to cancel, leaving only a real
part. That is, he proposed that:

(12)

We
proceed by cubing the two sides of Eq. (12):

(13)

To
solve for a and b, we set:

(14)

And we find that the solution is a = 2, and b = 1. So Bombelli showed that:

(15)

And therefore Eq. (11) is true because the imaginary parts
of the third roots cancel, leaving a real number.

The origin of imaginary numbers was in cubic
equations. These equations always have at
least one real root, clearly crossing the x-axis, yet del Ferro’s equation that
was supposed to give that root instead gave an expression that included square
root of negative numbers. Bombelli
showed that those “impossible number” were things that could be handled by introduction
of what we now call imaginary numbers. For
that accomplishment, there is a crater on the moon named after Rafael Bombelli.

Sunday, February 17, 2013

How does the brain evaluate a painful episode? When you look back at an unpleasant episode of
your life, how does your impression of it now relate to the actual experience
that you had during the episode?

Surprisingly, when we recall a painful experience we seem not
to evaluate it based on its duration, or its temporal integral, or its mean
pain. That is, it does not matter very
much if one experience was on average more painful than another, nor does it
matter that one experience was longer than another. Rather, we seem to evaluate the totality of a
painful experience using two factors: magnitude of the peak of the pain, and
the magnitude of the pain as the episode ended.
Here, I will describe the basic experiments that led to these ideas, and
then suggest a new interpretation of rather puzzling results regarding how the
brain evaluates effort in simple motor control tasks.

Cold water bath

In 1993, Kahneman and colleagues asked 32 volunteers at
University of California Berkeley to put both their hands in a cold water bath
for 5 seconds. Next, one hand was chosen
at random and placed in cold water for 60 seconds (or 90). After a brief rest period, the other hand was
placed in cold water for 90 seconds (or 60).
In these two episodes the temperature of the water was the same for the
first 60 seconds (21 degrees Centigrade).
However, in the last 30 seconds of the 90 second episode, the
temperature was increased by 1.1 deg. So
in the 90 second episode one hand always experienced a longer period of
discomfort, but the episode for that hand ended with slightly warmer
water.

During the time that their hand was in water the subjects
used their other hand to adjust a knob to continuously indicate their
discomfort. As you would expect, the
discomfort increased immediately as the hand was placed in the cold water,
reached a peak at around 60 seconds, and then declined for the next 30 seconds.

After the two episodes were completed, the subjects were
told that they would need to put their hand in cold water one more time but
that they could choose which episode they wanted. The main dependent variable was the subject’s
choice for this third episode. Logically, no one should pick the episode that lasted 90 seconds. But remarkably, most subjects (22 of 32, 69%)
preferred to repeat the longer episode.Indeed, most subjects indicated that the longer episode had caused less
overall discomfort!

This suggested that when people evaluate painful episodes,
what matters is not the duration, but rather the magnitude of the pain as the
episode ended. However, a potential
confound with the cold water experiment is that we know that memory fades with
time, and so perhaps evaluating the pain of an episode relies more on the ending
because the memory of the early parts have faded. Perhaps if the subjects were asked to
remember the episode a few days later, they would not recall it the same way as a
few minutes after the end of the episode.
Was this temporal decay the reason for the seemingly illogical choice? To test for this, Kahneman and colleagues
performed a new experiment.

The perceived pain of
a medical procedure

Redelmeier and Kahneman (1996) asked patients that were
undergoing colonoscopy (n=154) or lithotripsy (a procedure to destroy hardened
masses, n=133) to give assessment of their pain by pointing to a scale at one
minute intervals. The colonoscopy lasted
from 4-67 minutes, and the lithotripsy lasted from 18-51 min. One hour after the procedure the
patients were asked to judge the total amount of pain experienced using the
same scale.

To check for reliability of the evaluations, some of the
patients were asked to recall the experience 6 months (colonoscopy) or 1 year
(lithotripsy) later and again evaluate the total pain. The retrospective ratings at 6 months and 1
year were correlated at r=0.77 and r=0.54 for the two groups. For the colonoscopy group the ratings at 6
months had the same mean as at 1 hour, for the lithotripsy group the average
ratings at 1 year were 15% higher than at 1 hour.

In the colonoscopy procedure the pain intensity was higher
at start than at end, whereas in the lithotripsy procedure pain intensity was low
in the first few minutes and ended higher.

Having collected these data, the investigators asked what
aspect of the painful experience was a predictor of the immediate ratings at 1
hour, or the follow-up ratings at 6 months or 1 year. Duration of the procedure was not a predictor
of the immediate or follow-up ratings.
Rather, peak pain was the most powerful predictor of both ratings (r=0.6
for each), and end pain was the second most powerful predictor (r=0.4 for each). These correlations held for both of the
procedures. The combination of the two
factors increased the correlations to about 0.67 and 0.65 for immediate and
follow-up ratings.

So people’s impression of the relative pain they endured
during an episode remained fairly consistent at 1 hour and at many months after
the episode. Their impressions were predicted
by two aspects of their actual experience: magnitude of the peak of the pain,
and magnitude of the end pain. Duration
of the episode played little or no role. When we remember a painful episode, the most salient aspects of that
episode seem to be the peak of the pain, and how it ended. To improve our perception of a difficult episode, it may be
more beneficial to prolong it and gradually reduce the pain, rather than
shorten it and abruptly end the pain.

Perception of effort

This idea of peak-end perception of pain may help us
understand a rather puzzling result in the field of motor control. One of the fundamental questions in motor
control is how the brain evaluates effort.
The variables of interest are force and time, and the question is with regard
to our perception of effort as a function of these variables.

In 2004, +Konrad Kording, +Daniel Wolpert and colleagues performed an experiment in which volunteers held a robotic arm and experienced
a sinusoidal-like force profile of peak F and duration T.Next,
they experienced another force pattern of peak F’ and duration T’.They then asked their volunteers which force
they would like to experience again. They were told that they should choose the force that required the least effort. In
this way, the investigators estimated indifference curves, i.e., curves along
which the subjects were indifferent to changes in peak force and duration.

The rather unexpected result was that as the duration of a
force pattern increased (beyond about 200ms), the indifference curve also
increased. This means that given a
choice between some peak force and short duration, vs. the same peak force and longer duration, the subjects picked the longer duration! How could a longer duration of an effortful task be preferable to a shorter duration?

A close look at how the force patterns were produced
provides a possible answer. The forces
were sinusoidal with a period that depended on T. So as the duration increased, the rate at
which the force changed decreased. This
means that for a longer duration force, the forces gradually came to an end,
whereas for a short duration force, the forces rapidly came to an end. People preferred the gradually ending force,
despite the fact that they would be producing the forces for a longer amount of
time.

The peak-end hypothesis of pain perception may have
relevance to how the brain measures effort.

Acknowledgements: I am grateful to +Alaa Ahmed of
University of Colorado for discussions regarding these ideas.

Monday, January 21, 2013

In January of 2013, about a month after the horrific
shootings of children in Newtown, Connecticut, the Pew Research Center released
a survey of gun-related political leanings of people in America. They first asked the respondents to identify
themselves as either gun rights proponents, or gun control proponents. They then asked the respondents questions about their
political activity: did they contribute money to organizations that took a
position on gun policy? Had they
contacted a public official to express an opinion on gun policy? Had they signed a petition on gun
policy? Etc. The results indicated that those who
prioritized gun rights were 1.7 times more likely to have been politically
active (i.e., participated in one or more of these activities) than those who
prioritized gun control. Why should gun
rights advocates be almost twice as likely to be politically active than gun
control advocates?

To understand this behavior, it is useful to consider how
the human brain makes choices when faced with gains and losses.

In 1990, Kahneman and colleagues performed an experiment in
which they selected some participants and gave them a coffee mug as a gift. They then
asked them to assign a minimum price on the mug that they were willing to sell
it. These participants asked for about
$7. They then took another group of
participants and showed them the same mug and asked how much they would be
willing to pay to own it. They responded around $3. Knetsch (1989)
found that people who are given a chocolate bar want $1.83 to sell it, but will
pay only $0.90 to buy it. The difference
in the two prices is explained by loss aversion: the sellers evaluate the choice
of giving up something that they already own by viewing it as a psychological loss. In order to compensate for that loss, they request a lot of money. Buyers, on the other hand, evaluate the choice as a psychological
gain. They are willing to pay much less for the pleasure that they perceive in owning it.

In general, the pleasure that you feel if someone was to
give you an item tends to be much less than the pain you feel if you were to own
that item and were to lose it. This is
called an endowment effect.

Carmon and Ariely (2000) explain this behavior by suggesting
that when faced with loss of something (e.g., selling), people focus on their
sentiment toward surrendering the item (and not the money that they are
gaining), whereas when faced with gain of something (e.g., buying), people
focus on their sentiment toward what they forgo (typically money, and not the
item they are gaining).

Now let us consider the question of why gun rights
proponents are more politically active than gun control proponents. The current political climate is one in which
the President and the Congress are considering laws that would limit gun
rights. This is viewed as a loss to gun
rights proponents. In contrast, the same
laws are viewed as a gain for gun control advocates.

The gun rights proponents (but not the gun control proponents) are under the influence of the endowment effect because
if the proposed laws are enacted, it would result in a loss of what they already
‘own’. For them, the proposed laws carry
a negative psychological value. If we could generalize
from behavioral economics literature, we would speculate that this negative value is about twice as
large as the positive psychological value that would be gained from the
perspective of gun control proponents. This
may be the reason why the gun rights proponents are about twice as likely to be
politically active as the gun control proponents.

The deeper idea is that any change from the status quo will meet
with much stronger resistance by those who view the change as a loss, as
compared to the enthusiasm that it fosters in those who view the change as a
gain.

References

Carmon, Z. and Ariely, D. (2000) Focusing on the forgone:
How value can appear so different to buyers and sellers. Journal
of Consumer Research 30:15-29.

Kahneman D., Knetsch J., and Thaler R. (1990) Experimental tests
of the endowment effect and the coase theorem.
Journal of Political Economy
98:1325-1348.

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About Me

I was born in Iran and immigrated to the US at the age of 14. I was educated at Gonzaga University, University of Southern California, and finally MIT. I studied under the mentorship of Prof. Michael Arbib and Prof. Emilio Bizzi. I am currently Professor of Biomedical Engineering and Neuroscience, and the Director of the BME PhD Program at Johns Hopkins School of Medicine. I am a neuroscientist who uses mathematics to understand how the brain controls our movements.